Abstract
We consider the scalar sector of a general renormalizable theory and evaluate the effective potential through three loops analytically. We encounter three-loop vacuum bubble diagrams with up to two masses and six lines, which we solve using differential equations transformed into the favorable ϵ form of dimensional regularization. The master integrals of the canonical basis thus obtained are expressed in terms of cyclotomic polylogarithms up to weight four. We also introduce an algorithm for the numerical evaluation of cyclotomic polylogarithms with multiple-precision arithmetic, which is implemented in the Mathematica package cyclogpl.m supplied here.
Highlights
The effective potential [1,2] plays a very important role in investigating spontaneous symmetry breaking
The analysis of the effective potential at the two-loop order leads to the conclusion that the electroweak vacuum may be stable, critical, or slightly metastable up to very high energies of the order of the Planck scale [10,11,12]
An important step has recently been taken in Ref. [20], where the three-loop potential has been studied in a general renormalizable theory evaluating the loop integrals numerically [21,22,23]
Summary
The effective potential [1,2] plays a very important role in investigating spontaneous symmetry breaking. The analysis of the effective potential at the two-loop order leads to the conclusion that the electroweak vacuum may be stable, critical, or slightly metastable up to very high energies of the order of the Planck scale [10,11,12]. [20], where the three-loop potential has been studied in a general renormalizable theory evaluating the loop integrals numerically [21,22,23]. In the case of O(n) symmetry, we reproduce the known result for the scalar φ4 theory with spontaneous symmetry breaking. This theory is a matter of interest for the study of phase transitions, and its effective potential has been calculated in a series of papers [24,25]. The numerical evaluation of the cyclotomic polylogarithms that appear in our results is discussed in the appendix
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